Separate vs. Joint Continuity; A Tale of Four Topologies II
by
M. Henriksen
Harvey Mudd College
Coauthors: R. G. Woods (Univ of Manitoba)

Terminology and notation are as in Part I. Recall that a space in which every G_\delta is open is called a P-space. For any space (X, \alpha), the topology \alpha_\delta on X obtained by declaring each G_\delta to be open is the smallest P-space topology containing \alpha. Let \Delta(X) = { (x, x) in X² : x in X } denote the diagonal of X².

Theorem. If every point of X is a G_\delta , then \Delta(X) is a (closed and) discrete subspace of (X², \sigma), but no point of the diagonal of D = {0, 1}^c is isolated in ( \Delta(D), \sigma|_\Delta(D) ). On the other hand, \sigma|_\Delta(X) always contains \alpha_\delta, and if we let \Delta = C^* (X²) \cup { sp o (f ×f) : f in C(X) }, then \lambda_\Delta |_\Delta(X) = \alpha_\delta.

Theorem. If X is a nondiscrete P-space (indeed if X is zero-dimensional space with a nonisolated P-point), then \sigma is a strictly larger topology on X² than \lambda_H.

Theorem. If X and Y have cellular families of cardinality c, and c = |C(X)| = |C(Y)|, then |S(X ×Y)| > |C(X ×Y)|.

Theorem. If every nontrivial regular closed subset of X and Y has infinite boundary, then the Stone-Cech remainder of

Date received: April 12, 1996

Copyright © 1996 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caae-34.